Interest Rate Models: Vasicek and Hull-White

Short-rate models specify the instantaneous risk-free rate and derive bond prices and interest-rate derivatives from its future evolution. Vasicek provides the tractable mean-reverting Gaussian prototype; Hull-White adds a time-dependent drift so the model fits today’s entire discount curve exactly. Their simplicity makes them useful benchmarks, but their one-factor Gaussian structure limits smile and multi-curve realism.

Vasicek dynamics

Under the risk-neutral measure:

dr_t = a (b - r_t) dt + sigma dW_t

a controls mean reversion, b the long-run rate, and sigma rate volatility. The conditional distribution is Gaussian:

r_T | r_t ~ Normal(b + (r_t-b) exp(-a(T-t)),
                   sigma^2 (1-exp(-2a(T-t))) / (2a))

Zero-coupon prices are exponential affine:

P(t,T) = A(t,T) exp(-B(t,T) r_t)
B(t,T) = (1 - exp(-a(T-t))) / a

The affine form makes bond-option analytics and tree construction straightforward. The obvious defect is that a Gaussian rate can be negative; after the post-2014 negative-rate period, that is a feature for some currencies and a problem for others depending on product conventions.

Hull-White fits the curve

The one-factor Hull-White model generalizes Vasicek:

dr_t = [theta(t) - a r_t] dt + sigma(t) dW_t

Choose theta(t) to reproduce the initial discount curve P(0,T). For constant sigma, one representation is:

theta(t) = d f(0,t)/dt + a f(0,t)
           + sigma^2/(2 a^2) * (1 - exp(-2 a t))

where f(0,t) is the instantaneous forward rate. This separation is operationally valuable: the curve is matched exactly by construction, while a and sigma are calibrated to option prices.

Model	Fits initial curve exactly	Negative rates	Parameters
Vasicek	generally no	possible	a, b, sigma
Hull-White 1F	yes	possible	a, sigma, theta(t)
Hull-White 2F	yes	possible	two mean-reversion factors

Bond option pricing

Under Hull-White, a zero-coupon bond is lognormal under the relevant forward measure, giving a Black-style closed form for a bond call:

Call = P(0,T) N(h) - K P(0,S) N(h - sigma_P)
h = ln(P(0,T)/(K P(0,S))) / sigma_P + 0.5 sigma_P

Here S is option expiry, T bond maturity, and sigma_P is the model bond-price volatility integrated from 0 to S. Swaptions require a bond basket and are commonly priced with Jamshidian decomposition in one-factor models.

Calibration matters more than elegance

Calibrate to liquid swaptions or caps in the market’s normal/lognormal convention. A constant sigma generally cannot fit the expiry-maturity volatility matrix, so use piecewise-constant sigma(t) buckets:

def hw_theta(t, f0, dfdt, a, sigma):
    correction = sigma**2 / (2*a*a) * (1 - np.exp(-2*a*t))
    return dfdt(t) + a*f0(t) + correction

This computes the curve-fitting drift after the discount curve has been smoothed. Differentiating noisy bootstrapped forwards directly makes theta unusable; fit a differentiable curve representation or calculate integrated quantities without numerically differentiating noisy nodes.

Calibration should also distinguish fit from stability. Recalibrate on rolling quote snapshots and inspect how a one-tick bump changes a and each volatility bucket. A parameter surface that reprices today perfectly but produces discontinuous Bermudan deltas after a routine market move is not fit for hedging. Regularize adjacent volatility buckets and retain a benchmark calibration with frozen parameters for P&L attribution.

What one factor cannot hedge

A one-factor model moves every maturity through a single driver. Real yield curves exhibit level, slope, and curvature shocks; a 1F calibration can price an ATM swaption but misstate hedge ratios for Bermudans or callable bonds. Two-factor Hull-White helps, but neither model naturally fits a persistent volatility smile. Market models or stochastic-volatility extensions may be necessary for smile-sensitive books.

Also distinguish discounting from projection. Modern collateralized pricing uses an OIS discount curve and separate forwarding curves. A short-rate model for “the” rate does not excuse mixing curve identities; the yield-curve construction framework supplies those inputs.

Key takeaways

• Vasicek is tractable Gaussian mean reversion but generally misses the observed curve.
• Hull-White uses a time-dependent drift to fit the initial discount curve exactly.
• Bond-option analytics make both models strong benchmarks and calibration tools.
• Smooth the input curve before deriving time-dependent drifts.
• One-factor dynamics underrepresent curve-shape and volatility-smile risk.